arXiv subjects
Publications and source records attributed to Yu. Tschinkel.
Let $V_1$ be the Fano threefold given as a hypersurface of degree 6 in $P(1,1,1,2,3)$ (over a number field $K$). Then there exists a finite extension $K'/K$ such that the set of $K'$-rational points of $X$ is Zariski dense.
Let $X$ be an Enriques surface defined over a number field $K$. Then there exists a finite extension $K'/K$ such that the set of $K'$-rational points of $X$ is Zariski dense.
Let ${\cal L} = (L, \| \cdot \|_v)$ be an ample metrized invertible sheaf on a smooth quasi-projective algebraic variety $V$ defined over a number field. Denote by $N(V,{\cal L},B)$ the number of rational points in $V$ having ${\cal L}$-height $\leq B$. We consider the problem of a geometric and arithmetic interpretation of the asymptotic for $N(V,{\cal L},B)$ as $B \to \infty$ in connection with recent conjectures of Fujita concerning the Minimal Model Program for polarized algebraic varieties. We introduce the notions of ${\cal L}$-primitive varieties and ${\cal L}$-primitive fibrations. For ${\cal L}$-primitive varieties $V$ over $F$ we propose a method to define an adelic Tamagawa number $τ_{\cal L}(V)$ which is a generalization of the Tamagawa number $τ(V)$ introduced by Peyre for smooth Fano varieties. Our method allows us to construct Tamagawa numbers for $Q$-Fano varieties with at worst canonical singularities. In a series of examples of smooth polarized varieties and singular Fano varieties we show that our Tamagawa numbers express the dependence of the asymptotic of $N(V,{\cal L},B)$ on the choice of $v$-adic metrics on ${\cal L}$.